On Stresses of a Strip under Concentrated Loads at Longitudinal Borders

Abstract

The problem of plane elasticity on the stress state of strip S of constant width 2c whose opposite borders are loaded by concentrated forces is analyzed. An analytical solution to this problem is found via methods in the theory of functions of a complex variable. The stress components at an arbitrary point of the strip are defined in terms of two regular functions, Φ(z) and Ψ₁(z). To find these functions, the conformal mapping of domain S onto lower half-plane ζ is used. The half-plane problem is solved via classical technique based on Cauchy-type integrals. Exact analytical expressions for functions Φ(ζ) and Ψ₁(ζ) are obtained, which are then converted by the inverse conformal transformation into the required formulas for Φ(z) and Ψ₁(z). Since functions Ψ₁(z) and Φ'(z) were found to be related, the stresses in strip S were specified by function Φ(z) and its derivative Φ'(z). Graphs of normal and shear stresses on the lines parallel to the borders of the strip are presented. The stresses along the axis of the strip are compared to Filon’s data. The solution satisfies the differential equilibrium equations, boundary conditions, and the continuity equation.